Tim Chow,
[Re: Lucas-Penrose]
>In other words, they don't need to quantify over *all* consistent S.
>There's a section in "Shadows of the Mind" that talks about large
>cardinals, and Penrose doesn't claim that we're able to ascertain the
>consistency of arbitrarily powerful systems. It's just that according to
>Penrose, *if* S is unassailably reliable then Con(S) is unassailably
>reliable, so that the unassailably reliable truths can't be recursively
>axiomatizable.
We both agree that the principle I described (roughly, the mind has a
consistency oracle) is strong enough to get the conclusion Lucas/Penrose
want---the set of mathematical propositions that the mind can grasp as true
will be non-axiomatizable. Call this non-algorithmicity. We both agree that
the empirical evidence for the mind's having a consistency oracle is almost
non-existent.
But it seems to me that the principle you've given is much too weak for
non-algorithmicity. The principle
(Ref) If S is acceptable then Con(S) is too
is some sort of reflective closure condition. I agree with it, of course.
Relatedly, any theory of truth worth its salt will prove
If a set S of sentences is true, then Con(S) is true.
But how does reflective closure imply that "the unassailably reliable truths
can't be recursively axiomatizable"?
The principle you describe says that the mind's grasp of mathematical truth
is closed under some sort of reflection. I think that's very plausible, and
have argued in favour of it. Indeed, this is precisely the sort of
reflection that Feferman's 1991 work formalizes, yielding the notion of a
reflective closure Ref(S).
The point I wish to make is that if Penrose merely thinks that we can
reflect on what we accept, and see that we ought to accepts its consistency,
then I entirely agree. Similarly, as Tarski noted, if I accept
snow is white,
then I ought to accept
"snow is white" is true.
Furthermore, it seems to me that the analogous principle applies to
theories. If I accept PA, then I ought to accept "PA is true". What one
might call Tarskian reflection---the passage from a theory S to its
truth-theoretic extension Tr(S)---gives us this. For example, Tr(PA) implies
"All theorems of PA are true", and thus Con(PA). What one might call
Fefermanian reflection, the passage S |-> Ref(S), is much more powerful. For
example, the arithmetic content of Tr(PA) corresponds to ACA, which is
somewhat non-conservative over PA (ACA proves the local and uniform
reflection principles for PA). The arithmetic content of Ref(PA) corresponds
to a more powerful subsystem of second-order arithmetic.
But this does not imply that the set of propositions we shall in the end (in
some sense of "in the end") accept as a result of reflection is
non-axiomatizable. If we understand reflective closure in Feferman's sense,
then the relevant set is Ref(S), which is an axiomatic theory. It is S +
Kripke-Feferman truth axioms. So, I don't see how merely pointing out that
Con(S) is a reflective consequence of S helps to get the conclusion that the
mind is non-algorithmic. Even if I accept all the reflective consequences of
S, the result is still axiomatizable.
My interpretation of Lucas and Penrose is that it is this notion of
reflective consequence that is playing the central role, although they are
not clear about it. PA is part of what we already accept (modulo nominalism,
strict finitism, etc.). So, of course, we ought to accept Con(PA). Hence, G.
Penrose says that we have an "insight that goes beyond PA". True, but what
is this insight? So far as I can see, it is merely "PA is true". So, we pass
from (acceptance of) PA to (acceptance of) "PA is true". The passage from a
single sentence A to "A is true" is conservative. Amazingly, the
corresponding passage for theories is non-conservative. This is surprising:
PA doesn't imply G, but "PA is true" implies G. Truth-theoretic reflection
is non-conservative.
I'm suggesting that Lucas and Penrose may be mixing up two quite different
capacities which the human mind might possess:
(i) a capacity for reflective reasoning;
(ii) having a consistency oracle.
All agree that (ii)---having a consistency oracle---is sufficient for
non-algorithmicity. So is having a capacity to apply the omega-rule, or
something along those lines. But no one has given any empirical evidence for
the human mind's possession of such a capacity.
The notion (i) of reflective reasoning is important, and we have only begun
to scratch the surface. It is possible that the human capacity of reflection
is very powerful, and perhaps one might find an argument here for
non-algorithmicity. However, as far I presently understand the technical
situation, the capacity for reflection does not in itself imply
algorithmicity. It implies a certain kind of non-conservation. Reflection is
non-conservative, but it is not automatically non-algorithmic.
Best wishes --- Jeff
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Jeffrey Ketland
School of Philosophy, Psychology and Language Sciences
University of Edinburgh, David Hume Tower
George Square, Edinburgh EH8 9JX, United Kingdom
jeffrey.ketland at ed.ac.uk
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~